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Componentwise linear ideals

Published online by Cambridge University Press:  22 January 2016

Jürgen Herzog  and
Takayuki Hibi
Jürgen Herzog
Affiliation:
FB 6 Mathematik und Informatik, Universität-GHS-Essen, Essen 45117, Germany, juergen.herzog@uni-essen.de
Takayuki Hibi
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan, hibi@math.sci.osaka-u.ac.jp
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Abstract

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A componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal I Δ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 153 , 1999 , pp. 141 - 153
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

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